It is relatively easy to find the critical points of a system. Add that needs to be done is to set x' = 0 and y' = 0. The next step is to solve for x and y.

2. Determining the Jacobian Matrix

The Jacobian matrix of the system is

Hint: If the system isx' = Ax + Byy' = Cx + Dy

Then the system's critical point is (0, 0) and the Jacobian Matrix is

J = [ [A, B] [C, D] ]

3. Determine the Eigenvalues

As we recall the eigenvalues of a 2 × 2 matrix can be determined by solving the following characteristic equation:

λ^2 - trace * λ + determinant = 0

Where:* trace is the sum of the matrix's diagonal elements* determinant is the determinant of the matrix

The roots are crucial and determine the behavior of the critical point.

Types of critical points:* source: individual curve solutions x(t) and y(t) are trajectories going away from the critical point* sink: individual curve solutions x(t) and y(t) are trajectories go towards the critical point* saddle point: the critical point acts as a sink for some trajectories and a source for other trajectories* center: trajectories orbit around the critical point, most likely circular or elliptical orbits

Determining the type of critical points:* λ1 and λ2 are real and positive: critical point is a source. * λ1 and λ2 are real and negative: critical point is a sink* λ1 and λ2 are real and have opposite signs: critical point is a saddle* λ is a double root and it's positive: critical point is a source* λ is a double root and it's negative: critical point is a sink* λ = S ± Ti and S is positive: point is a source, trajectories are spiral* λ = S ± Ti and S is negative: point is a sink, trajectories are spiral* λ = ± Ti: critical point is a center